Generic Vopěnka’s Principle at YST2016

This is a talk at the Young Set Theory 2016 Conference in Copenhagen, Denmark, June 13-17, 2016.
Since this talk is similar to the one I gave a few weeks ago at the Rutgers Logic Seminar, I will just include here the abstract and links to other posts related to this work.

Abstract: Vopěnka’s Principle, introduced by Petr Vopěnka in the 1970’s, is the second-order assertion that for every proper class $\mathcal C$ of first-order structures in the same language, there are $B\neq A$, both in $\mathcal C$, such that $B$ elementarily embeds into $A$. In ${\rm ZFC}$, we can consider first-order Vopěnka’s Principle, which is the scheme of assertions ${\rm VP}(\Sigma_n)$, for $n\in\omega$, stating that Vopěnka’s Principle holds for $\Sigma_n$-definable (with parameters) classes. The principle ${\rm VP}(\Sigma_1)$ is a theorem of ${\rm ZFC}$; Bagaria showed that the principle ${\rm VP}(\Sigma_2)$ holds if and only if there is a proper class of supercompact cardinals, and for $n\geq 1$, ${\rm VP}(\Sigma_{n+2})$ holds if and only if there is a proper class of $C^{(n)}$-extendible cardinals, where $\kappa$ is $C^{(n)}$-extendible if for every $\alpha>\kappa$, there is an extendibility $j:V_\alpha\to V_\beta$ with $V_{j(\kappa)}\prec_{\Sigma_n} V$. We introduce Generic Vopěnka’s Principle, which asserts that the embeddings of Vopěnka’s Principle exist in some set-forcing extension. First-order Generic Vopěnka’s Principle is the scheme of assertions ${\rm gVP}(\Sigma_n)$ for $\Sigma_n$-definable classes of structures. The consistency strength of Generic Vopěnka’s Principle is measured by virtual large cardinals. Given a very large cardinal property $\mathcal A$, such as supercompact, $C^{(n)}$-extendible, or rank-into-rank, characterized by the existence of suitable set-sized embeddings, we say that a cardinal $\kappa$ is virtually $\mathcal A$ if the embeddings of $V$-structures characterizing $\mathcal A$ exist in some set-forcing extension. Unlike the similar sounding generic large cardinals, virtual large cardinals are actual large cardinals that fit between ineffables and $0^{\sharp}$ in the hierarchy. Remarkable cardinals introduced by Schindler turned out to be virtually supercompact. We show that ${\rm gVP}(\Sigma_2)$ is equiconsistent with a proper class of remarkable cardinals and for $n\geq 1$, ${\rm gVP}(\Sigma_{n+2})$ is equiconsistent with a proper class of virtually $C^{(n)}$-extendible cardinals. We conjecture that the equiconsistency results can be improved to get an equivalence. This is joint work with Joan Bagaria and Ralf Schindler.

@AsafKaragila I should have done a better job stating my question clearly. I wanted to know precisely what Yair answered: whether there are models where $\text{cf}(j(\kappa))$ is smaller than the size of $j(\kappa)$. Your interpretation of the question is very interesting as well, but I think much harder to answer.